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Knowledge Representation using First-Order Logic

Explore the application of First-Order Logic in knowledge representation. Learn about syntax, semantics, and common sense reasoning through examples and hands-on exercises. Dive into Predicate Calculus for enhanced expressive power in AI.

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Knowledge Representation using First-Order Logic

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  1. Knowledge Representation using First-Order Logic Reading: Chapter 8, 9.1-9.2 First lecture slides read: 8.1-8.2 Second lecture slides read: 8.3-8.4 Third lecture slides read: Chapter 9.1-9.2 (lecture slides spread across two class sessions) (Please read lecture topic material before and after each lecture on that topic)

  2. Review: KB |= S means |= (KB  S) • KB |= S is read “KB entails S.” • Means “S is true in every world (model) in which KB is true.” • Means “In the world, S follows from KB.” • KB |= S is equivalent to |= (KB  S) • Means “(KB  S) is true in every world (i.e., is valid).” • And so: {} |= S is equivalent to |= ({}  S) • So what does ({}  S) mean? • Means “True implies S.” • Means “S is valid.” • In Horn form, means “S is a fact.” p. 256 (3rd ed.; p. 281, 2nd ed.) • Why does {} mean True here, but False in resolution proofs?

  3. Review: (True  S) means “S is a fact.” • By convention, • The null conjunct is “syntactic sugar” for True. • The null disjunct is “syntactic sugar” for False. • Each is assigned the truth value of its identity element. • For conjuncts, True is the identity: (A  True)  A • For disjuncts, False is the identity: (A  False)  A • A KB is the conjunction of all of its sentences. • So in the expression: {} |= S • We see that {} is the null conjunct and means True. • The expression means “S is true in every world where True is true.” • I.e., “S is valid.” • Better way to think of it: {} does not exclude any worlds (models). • In Conjunctive Normal Form each clause is a disjunct. • So in, say, KB = { (P Q) (Q R) () (X Y Z) } • We see that () is the null disjunct and means False.

  4. Side Trip: Functions AND, OR, and null values(Note: These are “syntactic sugar” in logic.) function AND(arglist) returns a truth-value return ANDOR(arglist, True) function OR(arglist) returns a truth-value return ANDOR(arglist, False) function ANDOR(arglist, nullvalue) returns a truth-value /* nullvalue is the identity element for the caller. */ if (arglist = {}) then returnnullvalue if ( FIRST(arglist) = NOT(nullvalue) ) then return NOT(nullvalue) return ANDOR( REST(arglist) )

  5. Review: Resolution as Implication (OR A B C D) (OR ¬A E F G) ----------------------------- (OR B C D E F G) ->Same -> ->Same -> (NOT (OR B C D)) => A A => (OR E F G) ---------------------------------------------------- (NOT (OR B C D)) => (OR E F G) ---------------------------------------------------- (OR B C D E F G)

  6. Outline • Propositional Logic is Useful --- but has Limited Expressive Power • First Order Predicate Calculus (FOPC), or First Order Logic (FOL). • FOPC has greatly expanded expressive power, though still limited. • New Ontology • The world consists of OBJECTS (for propositional logic, the world was facts). • OBJECTS have PROPERTIES and engage in RELATIONS and FUNCTIONS. • New Syntax • Constants, Predicates, Functions, Properties, Quantifiers. • New Semantics • Meaning of new syntax. • Knowledge engineering in FOL • Required Reading: • For today, all of Chapter 8; for next lecture, all of Chapter 9.

  7. You will be expected to know • FOPC syntax and semantics • Syntax: Sentences, predicate symbols, function symbols, constant symbols, variables, quantifiers • Semantics: Models, interpretations • De Morgan’s rules for quantifiers • connections between  and  • Nested quantifiers • Difference between “ x  y P(x, y)” and “ x  y P(x, y)” •  x  y Likes(x, y) •  x  y Likes(x, y) • Translate simple English sentences to FOPC and back •  x  y Likes(x, y)  “Everyone has someone that they like.” •  x  y Likes(x, y)  “There is someone who likes every person.”

  8. Common Sense ReasoningExample, adapted from Lenat You are told: John drove to the grocery store and bought a pound of noodles, a pound of ground beef, and two pounds of tomatoes. • Is John 3 years old? • Is John a child? • What will John do with the purchases? • Did John have any money? • Does John have less money after going to the store? • Did John buy at least two tomatoes? • Were the tomatoes made in the supermarket? • Did John buy any meat? • Is John a vegetarian? • Will the tomatoes fit in John’s car? • Can Propositional Logic support these inferences?

  9. Exploring a Wumpus world If the Wumpus were here,stench should be here. Therefore it is here. Since, there is no breeze here, the pit must be there, and it must be OK here We need rather sophisticated reasoning here!

  10. Resolution example • KB = (B1,1 (P1,2 P2,1))  B1,1 • α = P1,2 P2,1 True! False in all worlds

  11. Pros and cons of propositional logic • Propositional logic is declarative -Knowledge and inference are separate  Propositional logic allows partial/disjunctive/negated information • unlike most programming languages and databases • Propositional logic is compositional: • meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2  Meaning in propositional logic is context-independent • unlike natural language, where meaning depends on context  Propositional logic has limited expressive power • E.g., cannot say “Pits cause breezes in adjacent squares.“ • except by writing one sentence for each square • Needs to refer to objects in the world, • Needs to express general rules

  12. First-Order Logic (FOL), also calledFirst-Order Predicate Calculus (FOPC) • Propositional logic assumes the world contains facts. • First-order logic (like natural language) assumes the world contains • Objects: people, houses, numbers, colors, baseball games, wars, … • Functions: father of, best friend, one more than, plus, … • Function arguments are objects; function returns an object • Objects generally correspond to English NOUNS • Predicates/Relations/Properties: red, round, prime, brother of, bigger than, part of, comes between, … • Predicate arguments are objects; predicate returns a truth value • Predicates generally correspond to English VERBS • First argument is generally the subject, the second the object

  13. Aside: First-Order Logic (FOL) vs. Second-Order Logic • First Order Logic (FOL) allows variables and general rules • “First order” because quantified variables represent objects. • “Predicate Calculus” because it quantifies over predicates on objects. • E.g., “Integral Calculus” quantifies over functions on numbers. • Aside: Second Order logic • “Second order” because quantified variables can also represent predicates and functions. • E.g., can define “Transitive Relation,” which is beyond FOPC. • Aside: In FOL we can state that a relationship is transitive • E.g., BrotherOf is a transitive relationship •  x, y, z BrotherOf(x,y)  BrotherOf(y,z) => BrotherOf(x,z) • Aside: In Second Order logic we can define “Transitive” •  P, x, y, z Transitive(P)  ( P(x,y)  P(y,z) => P(x,z) ) • Then we can state directly, Transitive(BrotherOf)

  14. FOL (or FOPC) Ontology: What kind of things exist in the world? What do we need to describe and reason about? Objects --- with their relations, functions, predicates, properties, and general rules. Reasoning Representation ------------------- A Formal Symbol System Inference --------------------- Formal Pattern Matching Syntax --------- What is said Semantics ------------- What it means Schema ------------- Rules of Inference Execution ------------- Search Strategy This lecture Next lecture

  15. Syntax of FOL: Basic elements • Constants KingJohn, 2, UCI,... • Predicates Brother, >,... • Functions Sqrt, LeftLegOf,... • Variables x, y, a, b,... • Connectives , , , ,  • Equality = • Quantifiers , 

  16. Syntax of FOL: Basic syntax elements are symbols • Constant Symbols (correspond to English nouns) • Stand for objects in the world. • E.g., KingJohn, 2, UCI, ... • Predicate Symbols (correspond to English verbs) • Stand for relations (maps a tuple of objects to a truth-value) • E.g., Brother(Richard, John), greater_than(3,2), ... • P(x, y) is usually read as “x is P of y.” • E.g., Mother(Ann, Sue) is usually “Ann is Mother of Sue.” • Function Symbols (correspond to English nouns) • Stand for functions (maps a tuple of objects to an object) • E.g., Sqrt(3), LeftLegOf(John), ... • Model (world)= set of domain objects, relations, functions • Interpretation maps symbols onto the model (world) • Very many interpretations are possible for each KB and world! • Job of the KB is to rule out models inconsistent with our knowledge.

  17. Syntax : Relations, Predicates, Properties, Functions • Mathematically, all the Relations, Predicates, Properties, and Functions CAN BE represented simply as sets of m-tuples of objects: • Let W be the set of objects in the world. • Let Wm = W x W x … (m times) … x W • The set of all possible m-tuples of objects from the world • An m-ary Relation is a subset of Wm. • Example: Let W = {John, Sue, Bill} • Then W2 = {<John, John>, <John, Sue>, …, <Sue, Sue>} • E.g., MarriedTo = {<John, Sue>, <Sue, John>} • E.g., FatherOf = {<John, Bill>} • Analogous to a constraint in CSPs • The constraint lists the m-tuples that satisfy it. • The relation lists the m-tuples that participate in it.

  18. Syntax : Relations, Predicates, Properties, Functions • A Predicate is a list of m-tuples making the predicate true. • E.g., PrimeFactorOf = {<2,4>, <2,6>, <3,6>, <2,8>, <3,9>, …} • This is the same as an m-ary Relation. • Predicates (and properties) generally correspond to English verbs. • A Property lists the m-tuples that have the property. • Formally, it is a predicate that is true of tuples having that property. • E.g., IsRed = {<Ball-5>, <Toy-7>, <Car-11>, …} • This is the same as an m-ary Relation. • A Function CAN BE represented as an m-ary relation • the first (m-1) objects are the arguments and the mth is the value. • E.g., Square = {<1, 1>, <2, 4>, <3, 9>, <4, 16>, …} • An Object CAN BE represented as a function of zero arguments that returns the object. • This is just a 1-ary relationship.

  19. Syntax of FOL: Terms • Term = logical expression that refers to an object • There are two kinds of terms: • Constant Symbols stand for (or name) objects: • E.g., KingJohn, 2, UCI, Wumpus, ... • Function Symbols map tuples of objects to an object: • E.g., LeftLeg(KingJohn), Mother(Mary), Sqrt(x) • This is nothing but a complicated kind of name • No “subroutine” call, no “return value”

  20. Syntax of FOL: Atomic Sentences • Atomic Sentences state facts (logical truth values). • An atomic sentence is a Predicate symbol, optionally followed by a parenthesized list of any argument terms • E.g., Married( Father(Richard), Mother(John) ) • An atomic sentence asserts that some relationship (some predicate) holds among the objects that are its arguments. • An Atomic Sentence is true in a given model if the relation referred to by the predicate symbol holds among the objects (terms) referred to by the arguments.

  21. Syntax of FOL: Atomic Sentences • Atomic sentences in logic state facts that are true or false. • Properties and m-ary relations do just that: LargerThan(2, 3) is false. BrotherOf(Mary, Pete) is false. Married(Father(Richard), Mother(John)) could be true or false. Properties and m-ary relations are Predicates that are true or false. • Note: Functions refer to objects, do not state facts, and form no sentence: • Brother(Pete) refers to John (his brother) and is neither true nor false. • Plus(2, 3) refers to the number 5 and is neither true nor false. • BrotherOf( Pete, Brother(Pete) ) is True. Binary relation is a truth value. Function refers to John, an object in the world, i.e., John is Pete’s brother. (Works well iff John is Pete’s only brother.)

  22. Syntax of FOL: Connectives & Complex Sentences • Complex Sentences are formed in the same way, and are formed using the same logical connectives, as we already know from propositional logic • The Logical Connectives: •  biconditional •  implication •  and •  or •  negation • Semantics for these logical connectives are the same as we already know from propositional logic.

  23. Complex Sentences • We make complex sentences with connectives (just like in propositional logic). property binary relation function objects connectives

  24. Examples • Brother(Richard, John)  Brother(John, Richard) • King(Richard)  King(John) • King(John) =>  King(Richard) • LessThan(Plus(1,2) ,4)  GreaterThan(1,2) (Semantics are the same as in propositional logic)

  25. Syntax of FOL: Variables • Variables range over objects in the world. • A variable is like a term because it represents an object. • A variable may be used wherever a term may be used. • Variables may be arguments to functions and predicates. • (A term with NO variables is called a ground term.) • (A variable not bound by a quantifier is called free.)

  26. Syntax of FOL: Logical Quantifiers • There are two Logical Quantifiers: • Universal:  x P(x) means “For all x, P(x).” • The “upside-down A” reminds you of “ALL.” • Existential:  x P(x) means “There exists x such that, P(x).” • The “upside-down E” reminds you of “EXISTS.” • Syntactic “sugar” --- we really only need one quantifier. •  x P(x)   x P(x) •  x P(x)   x P(x) • You can ALWAYS convert one quantifier to the other. • RULES:    and    • RULE: To move negation “in” across a quantifier, change the quantifier to “the other quantifier” and negate the predicate on “the other side.” •  x P(x)   x P(x) •  x P(x)   x P(x)

  27. Universal Quantification  •  means “for all” • Allows us to make statements about all objects that have certain properties • Can now state general rules:  x King(x) => Person(x) “All kings are persons.”  x Person(x) => HasHead(x) “Every person has a head.” • i Integer(i) => Integer(plus(i,1)) “If i is an integer then i+1 is an integer.” Note that • x King(x)  Person(x) is not correct! This would imply that all objects x are Kings and are People • x King(x) => Person(x) is the correct way to say this Note that => is the natural connective to use with  .

  28. Universal Quantification  • Universal quantification is equivalent to: • Conjunction of all sentences obtained by substitution of an object for the quantified variable. • All Cats are Mammals. • x Cat(x)  Mammal(x) • Conjunction of all sentences obtained by substitution of an object for the quantified variable: Cat(Spot)  Mammal(Spot)  Cat(Rick)  Mammal(Rick)  Cat(LAX)  Mammal(LAX)  Cat(Shayama)  Mammal(Shayama)  Cat(France)  Mammal(France)  Cat(Felix)  Mammal(Felix)  …

  29. Existential Quantification  •  x means “there exists an x such that….” (at least one object x) • Allows us to make statements about some object without naming it • Examples:  x King(x) “Some object is a king.”  x Lives_in(John, Castle(x)) “John lives in somebody’s castle.”  i Integer(i)  GreaterThan(i,0) “Some integer is greater than zero.” Note that  is the natural connective to use with  (And note that => is the natural connective to use with  )

  30. Existential Quantification  • Existential quantification is equivalent to: • Disjunction of all sentences obtained by substitution of an object for the quantified variable. • Spot has a sister who is a cat. • x Sister(x, Spot)  Cat(x) • Disjunction of all sentences obtained by substitution of an object for the quantified variable: Sister(Spot, Spot)  Cat(Spot)  Sister(Rick, Spot)  Cat(Rick)  Sister(LAX, Spot)  Cat(LAX)  Sister(Shayama, Spot)  Cat(Shayama)  Sister(France, Spot)  Cat(France)  Sister(Felix, Spot)  Cat(Felix)  …

  31. Combining Quantifiers --- Order (Scope) The order of “unlike” quantifiers is important.  x  y Loves(x,y) • For everyone (“all x”) there is someone (“exists y”) whom they love • y  x Loves(x,y) - there is someone (“exists y”) whom everyone loves (“all x”) Clearer with parentheses:  y (  x Loves(x,y) ) The order of “like” quantifiers does not matter. x y P(x, y)  y x P(x, y) x y P(x, y)  y x P(x, y)

  32. Connections between Quantifiers • Asserting that all x have property P is the same as asserting that does not exist any x that does not have the property P  x Likes(x, CS-171 class)   x Likes(x, CS-171 class) • Asserting that there exists an x with property P is the same as asserting that not all x do not have the property P  x Likes(x, IceCream)   x Likes(x, IceCream) In effect: -  is a conjunction over the universe of objects -  is a disjunction over the universe of objects Thus, DeMorgan’s rules can be applied

  33. De Morgan’s Law for Quantifiers Generalized De Morgan’s Rule De Morgan’s Rule Rule is simple: if you bring a negation inside a disjunction or a conjunction, always switch between them (or and, and  or).

  34. Aside: More syntactic sugar --- uniqueness • ! x is “syntactic sugar” for “There exists a unique x” • “There exists one and only one x” • “There exists exactly one x” • For example, E! x PresidentOfTheUSA(x) • This is just syntactic sugar: • ! x P(x) is the same as  x P(x)  ( y P(y) => (x = y) )

  35. Equality • term1 = term2is true under a given interpretation if and only if term1and term2refer to the same object • E.g., definition of Sibling in terms of Parent: x,ySibling(x,y)  [(x = y)  m,f  (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)] Equality can make reasoning much more difficult! (See R&N, section 9.5.5, page 353) You may not know when two objects are equal. E.g., Ancients did not know (MorningStar = EveningStar = Venus) You may have to prove x = y before proceeding E.g., a resolution prover may not know 2+1 is the same as 1+2

  36. FOL (or FOPC) Ontology: What kind of things exist in the world? What do we need to describe and reason about? Objects --- with their relations, functions, predicates, properties, and general rules. Reasoning Representation ------------------- A Formal Symbol System Inference --------------------- Formal Pattern Matching Syntax --------- What is said Semantics ------------- What it means Schema ------------- Rules of Inference Execution ------------- Search Strategy This lecture Next lecture

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